If P0 dollars are deposited in a bank account paying 5%...

If $P_{0}$ dollars are deposited in a bank account paying 5$\%$ interest compounded monthly, then the account has value $P_{0}\left(1+\frac{0.05}{12}\right)^{N}$ after $N$ months. Find, to the nearest integer $N,$ the number of months after which the account value doubles.

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Key Concepts

Doubling Time

Doubling time is the period required for a quantity undergoing exponential growth to double in size. In financial contexts, it represents the number of time intervals needed for an investment to grow to twice its original value, calculated using the compound interest formula and logarithms.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve equations where the unknown variable is an exponent. In the context of the problem, taking logarithms of both sides of the exponential equation helps isolate the variable representing the number of compounding periods.

Compound Interest

Compound interest is the process by which interest is added to the principal, so that from that moment on the interest that has been added also earns interest. In this problem, the interest is compounded monthly, meaning that the interest earned each month is added to the principal and the next month's interest is calculated on the new total.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant percentage over regular intervals. The compound interest formula is an example of an exponential growth model, where the account balance grows by the repeated multiplication of a fixed factor, reflecting the compounded nature of the interest.

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